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Perpetual emulation threshold ofHamiltonians-symmetric

To cite this article: D Trypogeorgos et al 2018 J. Phys. A: Math. Theor. 51 325302- The role of the geometric phases inadiabatic population tracking for nonHermitian Hamiltonians
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Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 51 (2018) 325302 (14pp)

https://doi.org/10.1088/1751-8121/aacc5e

Perpetual emulation thresholdof PT -symmetric Hamiltonians
D Trypogeorgos1, A Valdés-Curiel1, I B Spielman1
and C Emary2
1

Joint Quantum Institute, University of Maryland and National Institute of Standards
and Technology, College Park, MD 20742, United States of America
2
Joint Quantum Centre Durham-Newcastle, School of Mathematics, Statistics andPhysics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
E-mail: dtrypo@umd.edu
Received 4 June 2018
Accepted for publication 13 June 2018
Published 29 June 2018
Abstract

We describe a technique to emulate the dynamics of two-level PT -symmetricspin Hamiltonians, replete with gain and loss, using the unitary dynamics
of a larger quantum system. The two-level system in question is embedded
in a subspace of a four-level Hamiltonian, with the exterior levels acting as
reservoirs. The emulation time is normally finite, limited by the depletion of
the reservoirs. We show that it is possible to emulate the desired behaviourof the PT -symmetric Hamiltonian without depleting the reservoir levels, by
including an additional coupling between them. This extends the emulation
time indefinitely, when in the unbroken symmetry phase of the non-unitary
PT dynamics. We propose a realistic experimental implementation using
dynamically decoupled magnetic sublevels of ultracold atoms.
Keywords: PT -symmetry, non-Hermitian, pseudospins(Some figures may appear in colour only in the online journal)
1. Introduction
Quantum theory is our most successful description of nature. It describes the dynamics ofclosed systems by means of Hermitian Hamiltonians. The Hermitian constraint leads to unitary evolution with a real-valued, and hence measurable, eigenvalue spectrum. Remarkably,
this constraint is stronger than necessary: non-Hermitian Hamiltonians that commute with theOriginal content from this work may be used under the terms of the Creative
Commons Attribution 3.0 licence. Any further distribution of this work must maintainattribution to the author(s) and the title of the work, journal citation and DOI.
1751-8121/18/325302+14$33.00 © 2018 IOP Publishing Ltd Printed in the UK

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joint parity-time PT operator, also afford real-valued eigenvalues. These models emerged
from perturbative approaches in quantum field theory [1, 2] and can have all-real eigenvalues
in some parameter regimes [3, 4]. In such models, a free parameter of the Hamiltonian drivesa phase transition between two regimes where the PT symmetry is either unbroken or broken,
leading to real and complex eigenvalues respectively [5]. In the PT -symmetric phase a selfconsistent theory can be defined, with an appropriate redefinition of the inner-product of theHilbert space, which albeit violates the no-signaling principle [6].
Nevertheless, models based on PT -symmetric Hamiltonians are still useful in the study ofopen quantum systems since they can offer a simplified, alternative formulation to the Lindblad
equation description [7]. Generally, it is possible to derive a non-Hermitian Hamiltonian froma given Lindblad equation, that is a valid description of the systems’ dynamics until the probability of a quantum jump becomes too large.
Experiments in both the classical and the quantum domain have demonstrated
PT -symmetric systems [8, 9], e.g. using coupled optical waveguides [10–12], and driven systems [13–16]. Experiments in the quantum domain include the realisation of PT Hamiltonians
in exciton-polariton billiards [17, 18] and single-photon interferometers [19]. The physics ofthese systems becomes even richer close to the critical point between the broken and unbroken
regime; the phase associated with encircling the critical point is similar to the Berry phaseassociated with Dirac points in topological materials [20, 21].
In a series of papers [22–24], Wunner and coworkers showed that it is possible in principle to reproduce the dynamics of a two-level PT -symmetric Hamiltonian with a four-level
Hermitian system. In this configuration two of the levels map to the PT -symmetric levels
and the remaining two act as probability amplitude source and sink. The embedded PT -subspace is coupled to the rest of the system using time-dependent transition matrix elementsthat emulate the non-unitary particle flow between the subspaces; we refer to this scheme
as coherent emulation. Here we extend these works, initially by adopting a different calcul­
ational approach that permits analytic closed-form solutions in these four-level systems. Thisapproach makes it explicit that the emulation of the two-level PT -symmetric dynamics inthese works is always limited in time, and that it breaks down due to a depletion of the probability amplitude in the source level.
We then consider the case of periodic boundary conditions by including an additional coupling between source and sink levels, to counter the depletion of the source level. We showthat this cycling field increases the emulation time for any given set of parameters. Moreover,when the coupling strength of the cycling field exceeds a critical value, the duration of theemulation is extended indefinitely. Above this threshold the system dynamics dramaticallychange such that a genuinely periodic behaviour occurs. Implementation of this scheme
requires the solution of a pair of coupled, non-linear differential equations to pre-computethe time-dependent Hamiltonian required to emulate PT -symmetric behaviour in the target subspace. Our protocol therefore consists of a classical computation device to design acontrol sequence, different for all initial conditions, and a quantum system to implement thesequence. We use these results as the basis for the discussion of a realisation of the PT -symmetric dynamics in pseudospins corresponding to internal states of the groundstate manifoldof alkali atoms cooled to degeneracy.
2. Two-level PT -symmetric Hamiltonian
We focus on an extensively studied minimal model [11, 17, 25–28], that captures all the relevant information of such systems,
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−iΓ Λ
,
h
=
(1)
Λ iΓ

and aim to describe the full dynamics governed by h for all times and Γ, Λ > 0. The parity operator for this model is P = σx, where σi , i = x, y, z are the Pauli operators, and time
reversal gives T hT −1 = h∗. For this Hamiltonian, PT -symmetry implies Ph − h∗ P = 0 ,which√is manifestly obeyed by equation (1). The eigenvalues of h are ± = ±α/2, where
α = 2 Λ2 − Γ2 is the level splitting. This sets a natural energy scale for the system andallows us to use dimensionless time τ = αt ; coupling λ = 2Λ/α; and gain γ = 2Γ/α . We
assume here that we stay in the PT -unbroken phase such that α is real, i.e. Γ < Λ . As with
Hermitian Hamiltonians we define a time evolution operator G(τ ) that propagates the stateψ(τ ) = (ψ1 (τ ), ψ2 (τ ))T to time τ:G(τ ) = cos(τ /2)I − sin(τ /2)(γσz + iλσx ),(2)where I is the 2 × 2 unit matrix. We consider initial conditions with real-valued amplitudes
throughout ψ(0) = (cos(θ/2), sin(θ/2)).
Several aspects of this solution are of particular interest here. First, we consider the normn(τ ) ≡ |ψ1 (τ )|2 + |ψ2 (τ )|2 and the atomic magnetisation w(τ ) ≡ |ψ1 (τ )|2 − |ψ2 (τ )|2 .
Although our technique is valid for all initial states, for brevity of exposition we base oursubsequent discussion on the symmetric case when θ = π/2 . In this case, we obtainn(τ ) = λ2 − γ 2 cos(τ );(3)w(τ ) = −γ sin(τ ).
(4)For γ → 0, the norm becomes λ2 → 1, and the magnetisation disappears. In this limit, there
is no evolution since the initial state is an eigenstate of h. For θ = π/2 , oscillations in themagnet­isation only occur when h is non-Hermitian, and the oscillation frequency is givenby the splitting α. Also of interest is the relative phase between wave-function components
which, again for θ = π/2 , is defined bytan (Φ(τ )) = γλ(1 − cos(τ )).
(5)This phase difference is plotted in figure 1.
Two important timescales emerge from these equations. The first is the period of the
PT -symmetric oscillations in equation (4), which, with our choice of scaling for τ, is simply 2π . The second corresponds to important changes that take place at shorter times. These
are apparent in the phase difference in equation (5), plotted in figure 1. For small γ/λ , we
may approximate Φ(τ ) ≈ 2γ/λ sin2 (τ /2) and the dynamics unfold with the same period as
the magnetisation of equation (4). However, for increasing γ, the nonlinearity of the arctanfunction becomes important, and gives rise to sharp changes in the relative phase. Expanding
the phase about one of its minima, we find Φ(τ ) ∼ λγ(τ − τmin )2 /2 . Thus the characteristic√
timescale associated with these minima is τsharp = (λγ/2)−1/2 ∼ 2/γ  1 in the γ → λlimit. These two contrasting timescales set strict requirements for any experimental implementation of this scheme that become apparent when approaching the critical point fromeither the broken or unbroken symmetry phases. As such, we confine ourselves to looking into
the PT -symmetric phase only and explore how close to the critical point an experiment canapproach and still emulate the PT -dynamics faithfully.

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Figure 1. (a) The two populations p1 = |ψ1 |2 (black) and p2 = |ψ2 |2 (blue) of the√
PT -symmetric system with ψ1 (0) = ψ2 (0) = 1/ 2 and γ = 0.5λ. The norm n fromequation (3) is also shown. (b) The phase difference Φ of equation (5) as a function oftime for various γ. The time axis here is scaled with PT level-splitting α. Relative tothis scale, the minima of the phase become ever sharper as γ increases.

3. Emulation with a four-level system
Given that our understanding of Nature is in terms of Hermitian dynamics, we seek to create designer subspaces of larger systems that evolve according to a desired PT -symmetric
Hamiltonian. We simulate the non-unitary dynamics of equation (1) using a four-level systemwith wavefunction φ(τ ) = (φ0 (τ ), φ1 (τ ), φ2 (τ ), φ3 (τ )) T . We encode the PT dynamics inthe two central levels φ1, and φ2 , and use φ0 and φ3 as the sink and source levels. Our ‘emulator’ will thus be a system with dimensionless Hamiltonian3
δ0 (τ ) Ω01 (τ )0Ω03
Ω01 (τ )
0
λ
0 
.
H(τ ) = 
(6)
 0
λ
0
Ω23 (τ )
Ω030Ω23 (τ ) δ3 (τ )These couplings are sufficient to completely emulate the time-dynamics of PT -symmetric
Hamiltonians as in equation (1). The dimensionless detunings δ0 (τ ), δ3 (τ ) and the couplingsΩ01 (τ ), Ω23 (τ ) are time-dependent functions that we choose such that the behaviour of thetwo levels in the central subspace matches that of the PT -symmetric system, i.e. such that
φ1 (τ ) = ψ1 (τ ) and φ2 (τ ) = ψ2 (τ ). The static coupling Ω03 between the source level φ3 andsink level φ0 replenishes the source population by transferring particles from the sink level.
We initialise the system such that the population of the central subspace always starts as
ψ1 (0)2 + ψ2 (0)2 = 1; this implies a rescaling of φ1 , φ2 by an appropriate factor so that it isnormalised to unity at t  =  0.
We determine the dynamic detunings and couplings using the following procedure. We first
split the wavefunction components into real and imaginary parts, φi = φRi + iφIi , and use the
Schrödinger equation for our four-level system to get the equations:3

The corresponding physical Hamiltonian is H  (t) = 12 αH(αt).
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T

T
= H(τ ) φI0 , ψ1I , ψ2I , φI3 ;φ̇R0 , ψ̇1R , ψ̇2R , φ̇R3

T

T
φ̇I0 , ψ̇1I , ψ̇2I , φ̇I3 = −H(τ ) φR0 , ψ1R , ψ2R , φR3 ,(7)

where we have replaced φ1 and φ2 with their target wavefunctions ψ1 and ψ2, and whereφ̇i = dφi /dτ . We then write the second derivatives of ψ(τ ) with respect to time,ψ̈1R − Ω̇01 (τ )φI0 − Ω01 (τ )φ̇I0 − λψ̇2I = 0;(8)ψ̈2R − λψ̇1I − Ω̇23 (τ )φI3 − Ω23 (τ )φ̇I3 = 0,

for the real part and,

ψ̈1I + Ω̇01 (τ )φR0 + Ω01 (τ )φ̇R0 + λψ̇2R = 0;(9)ψ̈2I + λψ̇1R + Ω̇23 (τ )φR3 + Ω23 (τ )φ̇R3 = 0,

the imaginary part respectively. We use the equations for ψ̇1,2 in equation (7), together withequations (8) and (9) to solve for φ0,3 , and φ̇0,3 , which we then eliminate from the equations for φ̇0,3 in equation (7). From this set of four equations we obtain explicit expressionsfor δ0 (τ ) and δ3 (τ ) in terms of Ω01 (τ ), Ω23 (τ ) plus known quantities, together with a pair ofcoupled first-order differential equations2Ω̇01 = f1 Ω01 + f2 Ω301 + f3 Ω03 Ω−1
23 Ω01 ;(10)2Ω̇23 = g1 Ω23 + g2 Ω323 + g3 Ω03 Ω−1
01 Ω23 ,where fi ≡ fi (τ ) and gi ≡ gi (τ ) are functions defined solely in terms of φ1, φ2 and theirderivatives.
Our general solution procedure starts therefore by assuming initial values of the couplings
and solving the differential equations for Ω01 (τ ) and Ω23 (τ ). From these, we obtain δ0 (τ )and δ3 (τ ). Finally, we determine initial values of the source and sink levels, φ3 (τ = 0) andφ0 (τ = 0), by solving the equations for ψ̇1,2 at t  =  0. For simplicity in the following we willassume the initial couplings to be equal: Ω01 (τ = 0) = Ω23 (τ = 0) = Ωinit .
4. Open boundary conditions
While equation (10) must in general be solved numerically, in the absence of the cycling term
Ω03 = 0, they decouple and admit analytic solutions. Here we focus on initial conditions withequal amplitude in each of the target PT -symmetric states, giving2Ω23 (τ )λ2 − γ 2 cos τ + γ sin τ
(11)
,
=
Ωinit
1 − Ω2init (1 + λ2 τ /γ − cos τ − γ sin τ )andλ,δ(12)
0,3 (τ ) = 22
λ − γ cos τ ∓ γ sin τ

where Ω01 (τ ) = Ω23 (−τ ) (so that when time is reversed the source acts as a sink and viceversa), and δ0 (τ ) takes the upper sign and δ3 (τ ) the lower. The dynamic couplings are plotted for representative parameters in figure 2(a), making clear that the coupling Ω23 diverges.
Physically, this break down arises due to the depletion of source level, as shown in figure 2(b).
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Figure 2. (a) The two time-dependent couplings Ω01 (τ ) (black) and Ω23 (τ ) (blue)/π for parameters
required for emulation, as a function of dimensionless time τ√
γ = 0.5λ, Ω01 (0) = Ω23 (0) = Ωinit = 0.05λ, φ1 (0) = φ2 (0) = 1/ 2 and no cycling
field Ω03 = 0. (b) The corresponding sink and source populations p0 = |φ0 |2 (maroon)and p3 = |φ3 |2 (green). At time τ = 41.09π (indicated with the vertical dashed line),
the coupling Ω23 diverges as the source level is depleted. This breakdown is a common
feature of all solutions without periodic boundary conditions.

As the population of this level decreases, a stronger coupling Ω23 is required to maintain theconstant probability flux at rate γ into the system, and this necessarily stops when the level isdepleted.
Let us denote as τ ∗ the time at which Ω23 diverges, the maximum time we can expect ouremulation to run. We obtain τ ∗ as the smallest time at which the denominator in equation (11)vanishes, and plot it in figure 3(a) as a function of the starting coupling Ωinit for different
values of γ. For γ in the range 0.2λ  γ  0.8λ, changing γ does not alter the τ ∗ very much.
However, increasing γ above 0.8λ leads to a significant drop in maximum emulation time;
we observe similar behaviour for decreasing γ  0.2 . We can obtain a simple approximation
to the breakdown time by setting the oscillating terms in the denominator of equation (11) tozero. This gives
γ 1 − Ω2init
∗
(13)
τ ≈
,
(Ωinit λ)2valid for Ωinit  1. In the opposite regime, Ωinit  1, we obtain an alternative approximationτ ∗ ≈ γ/Ω2init ,(14)by expanding the denominator to first order in time, and setting the result equal to zero. Boththese expressions are good approximations in their respective regions of validity as shown infigure 3(a).
From the limits of equations (13) and (14), as well as from the full results in figure 3(a), we
see that the dominant behaviour of the breakdown time τ ∗ is an approximate scaling with Ω−2init .
Thus, in principle, we can always arrange our initial coupling strength to enable the emulation
to cover any time interval of interest. However, making Ωinit arbitrarily small is not without
cost: the initial values of the source and sink populations required by the emulation are
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Figure 3. The breakdown time τ ∗ as a function of the initial coupling strength Ωinit .

(a) Case without Ω03 = 0, for several values of γ/λ . The solid lines represent the exact
solutions; circles, the approximation of equation (13); and triangles, the approximationof equation (14). Lowering the coupling strength Ωint extends the duration of theemulation. (b) The breakdown time with periodic boundary conditions for a range ofcycling field strengths Ω03 and for fixed γ = 0.5λ. For large initial coupling strengthΩinit , the cycling field has little effect. As Ωinit is lowered, however, we see a dramatic
divergence of the breakdown time which corresponds to the onset of periodic behaviour.
Parameters not explicitly mentioned are the same as in figure 2.

−iγφI1 (0)iγφI2 (0)φ0 (0) =and φ3 (0) =
.
(15)
Ωinit
Ωinit

Since the total probability is not contained in the PT -symmetric subspace, we define thePT -symmetric fraction in terms of the populations pi = |φi |2 as
p1 (τ ) + p2 (τ )r(16)≡ min.
τ
i pi (τ )This expresses the minimum probability of finding the system in the PT -symmetric subspace.
This definition is motivated by an experimental resource limit since detecting the probability
distribution in all states becomes increasingly cumbersome as PT -symmetric fraction getssmaller. The PT -symmetric fraction is equal to r = min n(τ )/N0 , where n(τ ) is the normτof equation (3) and N0 = i pi (0) is the total population at time τ = 0, which is a conserved
quantity. Without they cycling field and with symmetric initial conditions, equations (3) and(15) imply
 2
λ − γ 2 cos(τ )1r(17)= min=2
τ
1 + (γ/Ωinit )1 + (γ/Ωinit )2 .
In the limit Ωinit /γ  1, this gives r ∼ Ω2init /γ 2 . Thus, decreasing Ωinit to extend the emulation
time necessarily leads to a corresponding decrease in the PT -symmetric fraction.

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5. Periodic boundary conditions and the perpetual emulation threshold
We now consider the effects of the cycling coupling field. In this case, equation (10) for Ω01
and Ω23 do not decouple and we obtain our results through numerical integration.
Figure 3(b) shows the breakdown time τ ∗ of the emulation as a function of the initialcoupling Ωinit for several values of the cycling strength. For large Ωinit , the cycling field does
not significantly affect the breakdown time. However, as Ωinit decreases, the breakdown timeincreases until it rapidly diverges at a value that increases with the cycling strength.
Figure 4 shows the populations of the sink and source levels as a function of time for several
values of Ω03 and for fixed initial coupling Ωinit = 0.05λ . As Ω03 is initially increased fromzero, the decline of p3 with time becomes shallower. The solution nevertheless terminates withp3  =  0, as in the Ω03 = 0 case. This trend continues until the critical value Ω03 = 0.014 12λ ofthe cycling strength is reached and the cycle-averaged gradients of p0 and p3 are both zero. At
this point, the behaviour of the populations is purely periodic; they never drop to zero and thecoupling Ω23 does not diverge. For yet larger values of Ω03 the populations reverse order suchthat p3 > p0, and, in the long-time limit, become periodic once again. The emulated PT -symmetric dynamics are correct for all times when Ω03 exceeds its critical value.
In figure 5 we explore the dynamics of the coupling functions and plot Ω01 (τ ) as a function of Ω23 (τ ). Without periodic boundary conditions (figure 5(a)) the trace simply diverges.
With Ω03 equal to its critical value (figure 5(b)) the functions form a single closed orbit which
illustrates the periodic nature of this solution. Then, for higher values of Ω03 (figure 5(c)), aftersome initial transients, the behaviour collapses onto a closed orbit. We note that these closed
orbits do not represent limit cycles because different initial values of the couplings give rise
to different oscillating states. Finally, in figure 5(d), we show results for a value of Ω03 far
in excess of its critical value. In this case, the long-time orbit develops a sharp kink feature,
which is a sign of the pronounced lack of time-reversal symmetry in the coupling functionsin this regime.
We have shown how the addition of a cycling field can elicit one of two distinct responses.
Below some critical coupling value, the emulation still terminates, albeit now with a larger
breakdown time cf. the case without periodic boundary conditions. Above the critical coupling, however, the cycling coupling field stabilises the dynamics, such that the emulationenters a purely periodic mode that can run indefinitely; a situation that only occurs whenγ = 0 in the absence of cycling. A clear boundary between the two regimes exists and isshown in figure 6 as a function of Ωinit and Ω03.
6. Experimental implementation and constraints
We can readily implement our scheme to coherently emulate a two-level PT -symmetric
Hamiltonian using ultracold atoms. Let us consider a 87Rb Bose–Einstein condensate in its
electronic groundstate 52S1/2. 87Rb is an alkali atom with nuclear spin I  =  3/2 so that the
groundstate splits into two hyperfine levels, F = 1, 2. We identify the PT -symmetric subspace with the |F, mF  = |1, −1 and |1, 0 levels and the reservoir levels with |1, 1 and |2, 0
levels. The |2, 0 level can be connected to the stretched levels of the F  =  1 manifold via two
microwave transitions that serve as the coupling to one of the reservoirs and the cycling field.
However, binary collisions between the two hyperfine manifolds might limit the lifetime of
the atomic cloud to 100 ms. For realistic rf coupling parameters this is in the ‘long emulation
time’ limit, and does not degrade the utility of this approach for simulating PT -symmetric
Hamiltonians. Alternatively, the continuous dynamical decoupling techniques described in8D Trypogeorgos et al

J. Phys. A: Math. Theor. 51 (2018) 325302

Figure 4. Populations p0 = |φ0 |2 (dashed) and p3 = |φ3 |2 (solid) as a function of time

for different values of 0  Ω03  0.05λ and other parameters as in figure 2. IncreasingΩ03 decreases the overall gradient of p3 until at the critical value Ω03 = 0.014 12λ thecycle-averaged values of Ω01 and Ω23 are constant and equal to one another. In thiscase we have purely periodic motion without decay. With further increase in Ω03, p3
becomes larger than p0 and in the long-time limit, we observe sustained oscillations of
the PT -symmetric subspace with no breakdown of the emulation.

Figure 5. Parametric plots of coupling functions Ω01 (τ ) versus Ω23 (τ ) over the time

range shown in figure 4. The four panels show results for different values of thecycling strength Ω03. (a) For Ω03 = 0, the trace is unbounded as Ω23 (τ ) diverges. (b)For Ω03 = 0.1412λ the motion is purely periodic. (c) Above this value, the behaviour
shows initial transients until once again a periodic trace is reached. Results are shownfor Ω03 = 0.05λ. (d) Finally, for strong cycling, Ω03 = 0.5λ, the driving field Ω01 (τ )becomes very strongly asymmetric in time. Parameters as in figure 2.

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Figure 6. Boundaries between oscillating solutions (above the curves, marked with anasterisk) and terminating solutions (below the curves) as a function of initial couplingΩinit and cycling strength Ω03 for γ/λ = 0.5, 0.9, 0.99 . Other parameters as figure 2.

[29, 30] allow this to be equally well realised all in the F  =  2 hyperfine manifold. We ignore
the non-linear interaction between the pseudospins that can be made negligible using lowdensity samples or an atomic Feshbach resonance; in any case, the non-linear interaction does
not qualitatively alter the behaviour of the system [23].
Ideally, we would like to be able to emulate the PT -symmetric system with arbitrary
parameters for as long a time as desired. While this is possible in principle, both with andwithout periodic boundary conditions, there exist a number of aspects inherent to this schemethat provide constraints on what is possible in practice.
The appearance of the timescale τsharp  1, discussed in section 2, represents a challenge
since for γ → λ the period of the population oscillations diverges, whilst τsharp remains fixed.
This timescale is directly reflected in the detunings. Expanding equation (12) about one of its(λ + γ) 1 − γ(λ + γ)(τ − τmin )2 /2 , which exhibits a timescale
maxima, we obtain δ0 (τ ) ≈ √−1/2. The corollary is that experimental control must be able
[γ(γ + λ)/2]∼ 2/γ = 2τsharp√
to simultaneously cover both small, 2/γ , and large, unity, time scales.
Imperfect preparation of the initial state has a similar effect in the faithful reproduction of
the PT -symmetric dynamics. Only the squared modulus of the wavefunction, i.e. the level
populations, needs to be taken into account when preparing the initial states. The required
phases can be absorbed into the driving fields through an appropriate gauge transformation,
which means that control parameters do not need to be recalculated for different initial phases.
An initial error of a few percent in state preparation of one of the PT -symmetric levels
induces a small discrepancy between the realised population dynamics and the target. Due tothe larger absolute population of source and sink levels, an error of a few percent in their prep­
aration will cause larger subsequent errors in the dynamics. This discrepancy, however, can bemade smaller if the preparation error is common-mode, where any offset in initial populationis shared by both source and sink levels. In any case, we can eliminate a state preparation error
with an initial weak measurement right after state preparation takes place. The weak measurement does not affect the initial populations, but provides the necessary information for recalibrating the dynamics of the emulation. This way, the time-dependence of the coupling fields
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and detunings is dictated by the measured initial state but the subsequent dynamics always
follow that of a PT -symmetric Hamiltonian albeit with slightly different parameters than theintended ones. Similarly, weak measurements performed while the dynamics take place canalso be used to correct for errors during the emulation.
Further constraints arise from the emulation scheme. The main practical constraint limiting the total emulation time without periodic boundary conditions is the fact that longer times
require larger populations of the source and sink levels, and this reduces the PT -symmetricfraction r. Small values of r demand a large dynamic range from the experimental detection
scheme. In practice, the dynamic range of the detectors sets the minimum signal-to-noise
ratio necessary for the chosen emulation parameters. This situation however can be improved
dramatically using periodic boundary conditions.
The equations in equation (7) used to determine the initial wave function components of the
PT -symmetric subspace at time τ = 0, are independent of the cycling strength Ω03. N0 andtherefore r are unaltered by the addition of the cycling field and equation (17) applies irrespectively of Ω03. Thus, the emulation times can be extended without changing the PT -symmetricfraction. The minimum value of r = rmin , set by experimental constraints, gives a minimum
value for the ratio:

Ωinit
rmin

,
(18)
γ
1 − rminthrough equation (17) and this applies independently of Ω03.
Figure 7 illustrates how this constraint can be taken into account to determine which values of Γ are experimentally accessible for a given rmin. Assuming we desire to simulate for a
fixed time of 10πα−1, say, we require a breakdown time τ ∗ > 10π. This requirement defines
an accessible region in the parameter space of Ωinit and Ω03; the solid lines in the figure 7
delineate the boundary of this region for various values of γ. Let us also assume a minimum
PT -symmetric fraction of rmin = 1/10 . The constraint equation (18) then also definesa region in the same parameter space, and these boundaries are plotted with dashed lines.
Clearly, for an emulation to both run long enough and have a sufficiently large r, we requirethat the two regions overlap. Whether this happens or not for a given γ can be seen fromfigure 7 as, for a feasible emulation, the two boundary lines will cross. This happens for
γ/λ = 0.5 for example, but not for γ/λ = 0.9. Indeed, from this plot we can obtain the maximum value of γ that it is possible to emulate. If we restricted ourselves to Ω03 /λ  2 such thatthe parameter range shown in figure 7 is all that is accessible, then we should be able to emulate the range 0  γ/λ  0.875. By way of contrast, we can use the analytical approximationof equation (13) to obtain the accessible range with open boundary conditions, and, for thesame parameters, this turns out to be 0  γ/λ  0.257. Use of periodic boundary conditions
therefore significantly enlarges the range of possible γ values that can be explored.
One further constraint comes from considering the change in size of the dynamic couplingsand detunings. This is not such a problem for the couplings Ω01 and Ω23 as these are made small
by construction to extend the simulation times. On the other hand, it is desirable for the detunings to also remain small but there is no a priori reason why this should be the case. Without
periodic boundary conditions, equation (12) shows that the detunings are always bounded as−λ − γ  δ0,3 (τ )  −λ + γ . These bounds change however, when we turn on the cycling
field. Figure 8 shows the size of the range of the detuning ∆δ0 ≡ |max(δ0 ) − min(δ0 )| as afunction of cycling strength Ω03.
The overall trend is that the range of values that the detunings take increases as we increase
the cycling strength. The same quantity for detuning δ3 shows little variation and is always
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J. Phys. A: Math. Theor. 51 (2018) 325302

Figure 7. The solid lines indicate the parameter values for which we obtain a simulation

break-down time τ ∗ = 10π for the indicated values of γ. In the regions above thesecurves, emulations last long enough to capture a minimum of ten Rabi oscillationsof the PT system. On the other hand the dashed lines indicate parameters for whichr = rmin = 1/10, which from equation (17) are the straight lines Ωinit = γ/3. Parameters
to the right of these lines lead to values of the PT -fraction r > rmin . To obtain feasible
emulations then, we require that these two lines cross for a given γ. For the parametersconsidered here (same as figure 2), this is the case for γ/λ = 0.5, 0.85, 0.875, but notfor γ/λ = 0.9.

Figure 8. The size of the detuning range ∆δ0 ≡ |max(δ0 ) − min(δ0 )| as a function ofΩ03 for various values of initial couplings Ωinit . Results are only plotted here for valuesof Ω03 that gives oscillatory solutions. The non-monotonic behaviour of these curves
arises because the detuning δ0 has multiple extrema which change order. Parameters
as in figure 2.

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J. Phys. A: Math. Theor. 51 (2018) 325302

order 2γ . Thus, to avoid requiring large detunings, we need to operate with as small Ω03 as
possible, in which case we obtain ∆δ0 ∼ 2γ . We note that although we show the range of δ0here, its increase is largely due to increase in magnitude of the minimum value, |min δ0 |.
7. Conclusions and outlook
Coherent emulation of non-Hermitian Hamiltonians can add to our understanding of thesemodels and their connection to open systems [9, 18]. In our scheme, a single unitary systemwith no intrinsic gain or loss emulates the dynamics of PT -symmetric Hamiltonians. The
system is first partitioned into two coherently coupled subspaces. Even though the couplings
are coherent, they can perfectly model particle gain and loss by appropriate manipulation of
their time dependence. The subspace of interest therefore mimics the evolution of an open system coupled to two reservoirs. Using periodic boundary conditions, with a non-zero coupling
between the two reservoirs, we are able to extend the emulation time indefinitely.
The coherent couplings between the embedded system and the ‘environment’ can becontrolled with standard experimental techniques [31]. This way, the effective decoherence and dynamic timescales can be adjusted arbitrarily to suit the experimental constraints.
Remarkably, decoherence slows down to effectively zero at the vicinity of the critical point[32], an effect which can be understood as a collapse of the Hilbert space to a single point andcan protect information stored in quantum memories [33].
The choice of emulation parameters dictates the properties of the reservoirs themselves.
This opens up exciting possibilities for engineering the correlation function of the reservoirs to
study (non)-Markovian open systems [34, 35]. These results can be extended to a wider classof non-Hermitian Hamiltonians using arbitrarily connected spin Hamiltonians as emulators.
Acknowledgments
We are thankful to P Solano for sparking our interest in the subject, and S Deffner for useful discussions. This work was initiated during DT’s stay at Newcastle, funded by a JuniorResearch Fellowship of the Joint Quantum Centre (JQC) Durham-Newcastle. In addition, thiswork was partially supported by the AFOSRs Quantum Matter MURI, NIST, and the NSF
through the PFC at the JQI.
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