PACS numbers: 42.50.-p; 03.65.-w; 42.82.-m · 2 to be written as ^a_ = ( i! )^a+ f^ (1) with properly chosen commutation relations between the harmonic oscillator mode and the noise

PT -symmetric photonic quantum systems with gain and lossdo not exist

Stefan Scheel and Alexander SzameitInstitut fur Physik, Universitat Rostock, Albert-Einstein-Straße 23, D-18059 Rostock, Germany

We discuss the impact of gain and loss on the evolution of photonic quantum states and findthat PT -symmetric quantum optics in gain/loss systems is not possible. Within the framework ofmacroscopic quantum electrodynamics we show that gain and loss are associated with non-compactand compact operator transformations, respectively. This implies a fundamentally different way inwhich quantum correlations between a quantum system and a reservoir are built up and destroyed.

PACS numbers: 42.50.-p; 03.65.-w; 42.82.-m

In 1998, Carl Bender challenged the perceived wisdomof quantum mechanics that the Hamiltonian operator de-scribing any quantum mechanical system has to be Her-mitian [1]. He showed that, in order to possess a realeigenvalue spectrum, a Hamiltonian does not have to beHermitian: also Hamiltonians that are invariant undercombined parity-time (PT ) symmetry operations havethis property [2, 3]. These findings had profound impactin particular on photonics research where the requiredpotential landscapes can be easily generated by appro-priately distributing gain and loss for electromagneticwaves. On this ground it was possible to show, for exam-ple, the existence of non-orthogonal eigenmodes [4], non-reciprocal light evolution [5], and PT -symmetric lasers[6, 7]. Even in fields beyond photonics PT -symmetry hasan impact, ranging from PT -symmetric atomic diffusion[8], superconducting wires [9, 10], and PT -symmetricelectronic circuits [11]. However, all these are classicalphenomena as only single electromagnetic wave packetsare involved. The experimental demonstration of trulyquantum features in PT -symmetric systems with gainand loss is still elusive. Here we show that this will remainto be the case. Our investigations unequivocally provethat the common approach for realising PT -symmetricsystems in photonics by concatenating lossy and ampli-fying media always results in thermally broadened quan-tum states. PT -symmetric quantum optics in gain/losssystems is therefore not possible.

PT -symmetric systems are described by a Hamilto-nian that is invariant under parity-time symmetry trans-formations [1]. In a more mathematical language thismeans that if the Hamiltionian H commutes with thePT -operator: [H, PT ] = 0 and the Hamiltonian and thePT -operator share the same set of eigenstates, then theeigenvalues of H are entirely real. A necessary condi-tion for this symmetry to hold is that the underlyingpotential obeys the relation V (−x) = V ∗(x). A com-monly used example is the complex anharmonic poten-tial V (x) = ix3. PT -symmetry is perceived as a complexextension of Hermitian quantum mechanics as it also pro-

vides a unitary time evolution [2]. Whereas complex po-tentials are difficult to realise in most physical systems,in 2007 it was shown that photonics provides a suitabletesting ground due to the complex nature of the refractiveindex [12, 13]. Since then, PT -symmetric systems havebeen explored in a variety of photonics platforms, rang-ing from waveguide arrays [4], fiber lattices [5], coupledoptical resonators [14], plasmonics [15] and microwavecavities [16]. Besides the broad range of platforms andobserved phenomena, all these systems make use of clas-sical electromagnetic waves. It it still an open questionas to whether or not quantised light shows the same be-haviour, although recent works imply that it might not assystems with a PT -symmetric Hamiltonian were shownto emit radiation [17, 18]. What we will unequivocallyshow is that indeed in all these platforms quantum opti-cal PT -symmetry does not exist.

The implementation of PT -symmetry in photonics isbased on the observation that the Schrodinger equationof quantum mechanics and the Helmholtz equation ofelectromagnetism are formally equivalent if the potentialV (x) in the Schrodinger equation is replaced by the re-fractive index profile n(x) in the Helmholtz equation [19].PT -symmetry then translates into the condition for thecomplex refractive index n(−x) = n∗(x), in particular,the real part nR(x) is symmetric and the imaginary partnI(x) is antisymmetric under the parity operation. Thelatter implies that loss in one propagation direction hasto be compensated by an identical gain in the oppositedirection [12]. Whereas this concept is well-defined forthe amplitudes of classical electromagnetic waves, this isno longer the case for the amplitude operators of quan-tum states of light as they have to obey certain commu-tation relations. For example, the amplitude operatorsfor a single harmonic oscillator have to fulfil the relation[a, a†] = 1 for all times. However, phenomenologically adissipation process is always accompanied by additional(Langevin) noise. Hence, the evolution equation for adamped harmonic oscillator mode with frequency ω has

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to be written as

˙a = (−iω − Γ)a+ f (1)

with properly chosen commutation relations between theharmonic oscillator mode and the noise operators, andwhere the fluctuation strength of the noise operator f isrelated to the damping rate Γ [20].

The appropriate framework in which to describe thepropagation of quantum states of light through absorb-ing and amplifying media is macroscopic quantum elec-trodynamics [21, 22]. Here the creation and annihila-tion operators of the free electromagnetic field have tobe replaced by new operators that describe the collectiveexcitation of the field and the absorbing or amplifyingmatter. Within the framework of linear response thistheory is exact. The result is a proper identification ofthe parameters Γ and f in Eq. (1) by phenomenologi-cal quantities such as absorption and transmission coef-ficients. This theory provides the basis for the propaga-tion of quantum states of light through absorbing andamplifying media [23, 24]. One first constructs a unitaryoperation in a larger Hilbert space of field and mediumoperators which, after projecting onto the field quantitiesalone, results in an effective, typically non-unitary evolu-tion of the quantum states of light [23]. Although the for-malism is very similar for absorbing and amplifying me-dia, there are crucial differences between them that im-pact the PT -symmetry. Viewing an optical element as afour-port device with two input and two output channelsfor light of a given frequency ω (note that in a linearlyresponding medium light modes of different frequenciesdo not mix), the quantum-state transformation at ab-sorbing media corresponds to a compact SU(4) transfor-mation [23] whereas the equivalent relation at amplify-ing media is a non-compact SU(2,2) transformation [24].This seemingly inoccuous difference has far-reaching con-sequences: an initial coherent quantum state |a0〉, afterpropagation through an absorbing medium with trans-mission coefficient T , remains a coherent quantum state,albeit with diminished coherent amplitude |Ta0〉. Onthe contrary, after propagation through an amplifyingmedium, a coherent state turns into a displaced thermalstate with an effective temperature that depends on thegain (for details of the calculation, see SupplementaryMaterial).

We illustrate this fundamental difference by the prop-agation of a coherent quantum state through a systemthat consists of concatenated regions of loss and gain.In Fig. 1 we show the Wigner function (a phase-spacedistribution function that is formally equivalent to thequantum state [25]) of a coherent quantum state withcoherent amplitude a0 = 3 + 3i (left), after transmissionthrough an optical device with transmission coefficientT = 2/3 (center), and after propagation through a gainmedium with G = 1/T (right). One clearly observes that

initial and final states after propagation through mediawith loss and subsequent gain are not equivalent.

neutral loss neutral gain neutral

01

23

45

Re(a) 0

1

2

3

4

5

Im(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

01

23

45

Re(a) 0

1

2

3

4

5

Im(a) 01

23

45

Re(a) 0

1

2

3

4

5

Im(a)

W

Win

Wloss

Wgain

FIG. 1: Evolution of a coherent state through a concatenatedsystem of loss and gain. Shown on the left is the initial Wignerfunction of a coherent state with a0 = 3 + 3i, in the centerafter transmission through an optical device with transmis-sion coefficient T = 2/3, and on the right after propagationthrough a gain medium with G = 1/T . It can be clearlyobserved that the state significantly changes.

Despite the apparent simplicity of our example, it doeshave far-reaching consequences for any attempt to studyPT -symmetry of quantum optical systems. First, gainalways adds thermal noise to a quantum state, no mat-ter how gain and loss are spatially distributed. Second,we have chosen coherent quantum states that are, on theone hand, minimum-uncertainty states that closely re-semble classical states [25] and, on the other hand, havethe unique property that their purity is not affected byloss. Any other type of quantum state will already bedrastically altered by an absorbing medium. For exam-ple, pure photon-number states turn into mixed stateswith lower photon numbers [23]. As a consequence, anyquantum state of light is crucially altered when propa-gating through any distribution of gain and loss. In otherwords, in a concatenated gain/loss system the quantumeigenstates of a Hamiltonian can never be eigenstates ofthe PT -operator, such that for quantum states there isalways [H, PT ] 6= 0. This brings us to the conclusionthat in such systems PT -symmetric quantum optics doesnot exist.

The reason behind this surprising result is the factthat, in order to obtain amplification, the quantum sys-tem under study has to be coupled to an external reser-voir that provides the necessary energy input. This cou-pling necessarily introduces noise that can be cast into aform similar to Eq. (1),

˙a = (−iω + Γ)a+ f† , (2)

where the (Langevin) noise operators fulfil the same com-

3

mutation relations as before. The crucial difference be-tween loss and gain is the way in which the external reser-voir is coupled to the quantum system (see Fig. 2). Incase of gain, the noncompact SU(2,2) group transforma-tion implies a build-up of quantum correlations betweensystem and reservoir that, when only observing the sys-tem, are destroyed and manifest themselves as thermalfluctuations. This is a similar mechanism as that ob-served in two-mode squeezing (which is described by aSU(1,1) transformation) where the quantum correlationsof the two squeezed modes result in thermal distributionsof the individual modes. Therefore, there is no quantumgain mechanism that can compensate for any quantumloss process.

non-hermitian

hermitian

field

loss gainf

reservoir 1 reservoir 2

f†

FIG. 2: Photonic quantum systems with gain and loss. Inorder to describe a non-Hermitian quantum system with gainand loss, it has to be coupled to external reservoirs that actas source and sink, respectively. This coupling introducesLangevin noise f and f† to ensure the Hermiticity of thefull scheme. This noise, however, always alters propagatingquantum states, even the eigenstates of the Hamiltonian ofthe quantum system.

Importantly, our result does not only hold for harmonicoscillator modes such as photons. Indeed, any systemin which a complex potential is derived from a couplingto a reservoir and, accordingly, admits a description bya Langevin equation suffers from a similar conclusion.Hence, our conclusions are not restricted to bosonic sys-tems or, indeed, photons, but also hold for fermionic sys-tems, such as electrons.

To summarize our work, we have shown that PT -symmetric quantum optics is not possible within the cur-rent perception of implementing PT -symmetry, that is,using gain and loss. However, we foresee three alternativeapproaches that might allow the observation of phenom-ena arising from PT -symmetry in the quantum realm.First, one would have to realize a complex PT -invariantHamiltonian without coupling the quantum system toan external reservoir. However, to date such a conceptis elusive. Second, one would need to construct noise-

less amplifiers. Yet, for deterministic gain processes thisviolates the No-cloning theorem of quantum mechanics[26]. Indeed, relaxing the determinism constraint andallowing probabilistic processes may result in devisingprobabilistic noiseless amplifiers [27]. An altogether dif-ferent route involves replacing the active gain by yet an-other passive loss medium such that the overall systemis lossy, which results in so-called passive PT -symmetricsystems. Such structures are already implemented [28–30] and seem promising candidates for observing physicsakin to PT -symmetric quantum optics.

The authors thank Mark Kremer for helping to preparethe figures and acknowledge the Deutsche Forschungsge-meinschaft (grant BL 574/13-1) for financial support.

Appendix

The idea of our approach is to discretize the time evo-lution of the electromagnetic field and construct input-output relations between photonic amplitude operatorsbefore and after the propagation through an optical de-vice [24]. Let the amplitude operators of the radiationfield at frequency ω at the input of the optical device bedenoted by a, the corresponding output amplitude op-erators by b, and the (Langevin) operators associatedwith the device by g. Then, the following input-outputrelations read b = Ta + Ad where the transformationand absorption matrices satisfy TT+ + σAA+ = I andσ = +1, d = g for absorption and σ = −1, d = g† foramplification.

Although a unitary evolution of the field operatorsalone is no longer possible, one can nevertheless con-struct a unitary evolution of the combined field-devicesystem. Define the four-vector operators α = (a, d)T

and β = (b, f)T where f = h for absorption and f = h†

for amplification with some auxiliary bosonic device vari-ables h. Then, the input-output relations can be elevatedto a unitary relation between the four-vector operatorsas β = Λα with

ΛJΛ+ = J , J =

(I 00 σI

).

If one introduces the commuting positive Hermitian ma-

trices C =√

TT+ and S =√

AA+, then the unitarymatrix Λ can be written as [24]

Λ =

(T A

−σSC−1T CS−1A

).

The input-output relation for the amplitude operatorscan be cast into a quantum-state transformation formula.Let the density operator of the input quantum state begiven as a functional of the amplitude operators α andα†, ρin = ρin[α, α†], then the transformed quantum stateat the output is ρout = ρin

[JΛ+Jα,JΛTJα†

]. Taking

4

the partial trace over the device variables laeves one withthe quantum state of the radiation field alone.

The equivalence between density operators and quasi-probability functions implies a similar transformationrule for the phase-space functions. However, care needsto be taken as the SU(2,2) transformation associated withgain mixes creation and annihilation operators, exceptfor the Wigner function associated with symmetric oper-ator ordering for which Wout(α) = Win (JΛ+Jα) holds.

The above relations can now be used to constructquantum states after propagation through lossy and am-plifying media. The simplest example is a coherent state|a0〉 whose Wigner function is given by the GaussianW (a) = 2

π exp(−2|a− a0|2

)with obvious generalization

for multimode states. At lossy devices, a two-mode co-herent state |a〉 results in a Wigner function Wout(a) =∫d2gWout(α) =

(2π

)2exp

(−2|a−Ta0 −Ag0|2

)which

again represents a coherent state |Ta0 +Ag0〉. If we takethe device to be initially in its vacuum state, g0 = 0, weare left with a coherent state |Ta0〉.

In the case of gain, a lengthy but straightforward ap-plication of the quantum-state transformation relationsshows that the same coherent state transforms into

Wout(a) =

(2

π

)21

det(2TT+ − I)×

× exp[−2(a+ − a+

0 T+)

(2TT+ − I)−1 (a−Ta0)]

which is no longer a coherent state, but a displacedthermal state whose temperature depends on the gain.Neglecting reflection at the interface of the device, the(single-mode) Wigner function of the transmitted lightis

Wout(a) =2

π

1

2|T |2 − 1exp

[−2|a− Ta0|2

2|T |2 − 1

]where the transmission coefficient |T | > 1 due to gain.

If we now construct a hypothetical device that con-sists of a sequence of a lossy medium with transmissioncoefficient |T | < 1 followed by a gain medium with trans-mission coefficient |G| = 1/|T | > 1, this would mimica PT-symmetric system. However, as we have seen,noise enters both during the absorption as well as theamplification process. In fact, starting with a (single-mode) Wigner function Win(a) = 2

π exp(−2|a− a0|2

),

after propagation through a lossy medium this turns intoWloss(a) = 2

π exp(−2|a− Ta0|2

). Reversing the loss by

amplification then results in a Wigner function

Wgain(a) =2

π

1

2|G|2 − 1exp

[−2|a− a0|2

2|G|2 − 1

]which is a (thermally) broadened version of the originalWigner function with the mean thermal photon num-ber nth = |G|2 − 1 or, equivalently, Teff = − ~ω

kBln(1 −

|T |2). What it also shows is that only the first-order

moments of the amplitude operators are conserved bythis system, not even the second-order moments. In-deed, the mean number of photons contained in a quan-tum state with Wigner function Wgain(a) is 〈n〉 =∫d2a

(|a|2 − 1

2

)Wgain(a) = |a0|2 + nth, which deviates

from the coherent state result by the addition of the meanthermal photon number associated with the gain.

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