Wading through the Hilbert space

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<center><wz tagtotip="kaupptip">[[File:kaupp16a-g2.png|400px]]</wz></center>
 
<center><wz tagtotip="kaupptip">[[File:kaupp16a-g2.png|400px]]</wz></center>
<span id="kaupptip">$Taking something into account relaxes it to g^{(2)}\lesssim0.7$.</span>
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<span id="kaupptip">Taking background into account relaxes it to $g^{(2)}\lesssim0.7$.</span>
  
 
You also hear it quite a lot in conferences, which is more difficult to reproduce, although you see it in posters too, e.g., this is from Heindel ''et al.'' [http://www.treasure-project.eu/wp-content/uploads/2011/09/HeindelPoster.pdf]:
 
You also hear it quite a lot in conferences, which is more difficult to reproduce, although you see it in posters too, e.g., this is from Heindel ''et al.'' [http://www.treasure-project.eu/wp-content/uploads/2011/09/HeindelPoster.pdf]:

Revision as of 14:40, 6 June 2019

The Hilbert space is a big place. In the words of Douglas Adams (?!):

Bigger than the biggest thing ever and then some. Much bigger than that in fact, really amazingly immense, a totally stunning size, real 'wow, that's big', time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we're trying to get across here.

We think we know it well because its canonical basis makes it look easy. Every member of the space comes in the form:

$$\sum_{k=0}^{\infty} \alpha_k\ket{k}$$

Really easy. The $\alpha_k$ are complex and they have to be normalized, $\sum_{k=0}^\infty|\alpha_k|^2=1$, but this is just basic linear algebra stuff.

That is actually deceiving as we're not quite gifted in imagining all the structure that may arise from an infinite set of complex numbers. When we think of quantum states, we think in terms of families. The simplest ones are those that constitute the canonical basis, the Fock states $\ket{k}$. That's one photon for you: $\ket{1}$.

Then, since Glauber, we know of coherent states $\ket{\alpha}$, Poissonian superpositions of photons, and thermal (also called "chaotic") states, with exponentially decaying probabilities from the vacuum. Thermal states are also statistical mixtures but that doesn't matter too much. Oh, and I forgot vacuum $\ket{0}$ although we mentioned it already. That's already quite of a good collection. I'm sure I wasn't the first but in 2006 I introduced mixtures of thermal and coherent, that I called, rather loosely, "cothermal states" Effects of Bose-Einstein condensation of exciton polaritons in microcavities on the polarization of emitted light. F. P. Laussy, I. A. Shelykh, G. Malpuech and A. Kavokin in Phys. Rev. B 73:035315 (2006). Pdf-48px.png. You can also interpolate between Fock and coherent states Glauber--Sudarshan P representation of negative binomial states. K. Matsuo in Phys. Rev. A 41:519 (1990).. I don't know of interpolations between Fock and thermal states but this is certainly not difficult to work out. There is also the big family of squeezed and Gaussian states.

Still, all these states are far from providing even a small portion of what populates the whole Hilbert space, even though they are the most important instances.

Why would you care? Besides curiosity, a first reason came from our exciting with quantum light line of research. When you excite something with any of the funny new sources that we develop, you typically bring your target in a quantum state that is not a member of the family above (showing again how it fails to be comprehensive; our first exploration already brought us away from the lobby of the Hilbert hotel).

Take the simplest source, the single photon source (SPS). How does a quantum harmonic oscillator swing when you drive it with a SPS? It's not a Fock state $\ket{1}$. What is it? (it's easy, compute it). But more than that, what could it be? What states can we drive with a SPS? What states can we drive with any source at all? What states are in fact possible?

With Eduardo Zubizarreta Casalengua, a Master student (who got most of these results even before starting his master), we worked out in detail a new charting of the Hilbert space, that we had introduced earlier Excitation with quantum light. I. Exciting a harmonic oscillator. J. C. López Carreño and F.P. Laussy in Phys. Rev. A 94:063825 (2016). Pdf-48px.png. The rulers are the Glauber correlators, $n_0$ (the mean population) and $g^{(n)}$, the $n$th order coherence function:

$$g^{(n)}\equiv\langle a^{\dagger n}a^n\rangle/\langle\ud{a}a\rangle^n\,,$$

where $a$ is the annihilation (ladder) operator.

This is how the Hilbert space for up to 3 particles (we call it $\mathcal{H}_3$) looks like:

Sahs-H3.png

(we address $\mathcal{H}_2$ and some generic features of $\mathcal{H}_N$ in the text.) Since we have three particles at most, $g^{(4)}$ and all higher $g^{(n)}$ are here zero, so we can move in a 3D space. What this shows is the boundary for the allowed physical states (that exist for a given $(n_0, g^{(2)}, g^{(3)})$. Everything above this circumvoluted surface is forbidden, everything below till the bottom plane (set by $n_0=0$) is allowed. This is an infinite volume as although $n_0\le 3$, both $g^{(2)}$ and $g^{(3)}$ can take arbitrary values. In addition to the sole boundary, we have also worked out the density of states $\mathcal{P}_g \left( n_0, g^{(2)},g^{(3)}\right)$ populating the Hilbert space charted in this way, which is indicated on the surface only through the colour (which could also be extended to the space below, which we don't do here for clarity). The expression for the density of probability is remarkably simple (you can find it in the text, Eq. (25)) and depends only on the population... but it is defined in a complex geometrical space. Considering projections allows to get a better grip of how the states are distributed:

Sahs-project-H3.png

If you pick up a state randomly that has at most 3 particles, i.e., of the type $\alpha_0\ket{0}+\alpha_1\ket{1}+\alpha_2\ket{2}+\alpha_3\ket{3}$, these are the chances that you will find it with such and such a population and particle fluctuations. Importantly, there are some limits to the allowed values you will get. Namely, your state has to satisfy:

$$\max\left( n_0 g^{(3)},\frac{n_0 g^{(3)}}{3}+\frac{2}{n_0}-\frac{2}{n_0^2}\right) \leq g^{(2)}\leq\frac{n_0g^{(3)}}{2} + \frac{1}{n_0}\,,$$

and

$$\frac{2 g^{(2)}}{n_0}-\frac{2}{n_0^2}\leq g^{(3)}\leq\min\left(\frac{g^{(2)}}{n_0},\frac{3 g^{(2)}}{n_0}-\frac{6}{n_0^2}+\frac{6}{n_0^3} \right)\,,$$

as well as a condition on $n_0$ which is however too big to be written there (cf. Eqs. (28,29,35—36) of the text). As you see, your particle fluctuations are bounded both from above and below. So it's not like you can go anywhere. Granted, this is for a truncated space, and the lack of enough particles could understandably cramp the span of your reach. Note however particular that even if you allow a maximum of 3 particles, you can get arbitrarily high superbunching, by allowing $n_0$ to be small (note the $1/n_0$ divergence on the rhs).

Spectacle.w13805.png

Anyway, it is true that adding more particles wins more space. But not all the space. These are the boundaries for arbitrary $N$ (the case $N=3$ gives the conditions for the lhs and rhs of the above inequalities to be valid as well, as not all combinations of $n_0$ and $g^{(2)}$ or $g^{(3)}$ are allowed):

$$ \frac{\lfloor n_0 \rfloor !}{(\lfloor n_0 \rfloor-k)!n_0^k} \left(1+ \frac{k(n_0-\lfloor n_0 \rfloor)}{\lfloor n_0 \rfloor+ 1 - k} \right) \leq g^{(k)} \leq \frac{(N-1)!}{(N-k)!} \frac{1}{n_0^{k-1}}$$

From this expression, one sees that with increasing $N$, the rhs becomes unbounded, i.e., with enough particles, you can get $g^{(k)}$ as large as you want for any population $n_0$ and order $k$ of the correlation. However, the lhs does not depend on $N$, and while $n_0$ can now take larger values (up to $N$), it cannot access smaller values of $g^{(k)}$. This is how increasing portion of the $(n_0,g^{(2)})$ space get covered with increasing $N$ (a similar result holds for $(n_0,g^{(k)})$):

Sahs-hilbertcat.png

The white area (where Schrödinger's cat is playing) is forbidden, there are no states in the Hilbert space providing such combinations of populations and intensity fluctuations.

As an illustration that this relief of the Hilbert space is not familiar even to those who are used to cruise its waters, we have highlighted the area of possible states such that $g^{(2)}<1/2$ and $n_0>1$. That is, states with more than one particle on average, but an antibunching smaller than 0.5. The point is that $g^{(2)}<0.5$ has been "accepted" as a criterion for single-photon sources or single-photon states, the logic being that since $\ket{2}$ has a $g^{(2)}$ of 0.5, anything with a smaller $g^{(2)}$ has to have one particle at most. This is badly broken logic (since the $g^{(2)}$ should not then be merely smaller than 0.5, but exactly 0). Going deeper into this question is the topic of another work, our Criterion for single-photon source, and I will postpone further discussion about this too when the paper will be published, although you can already consult the work itself, arXiv:1610.06126. Instead, as a conclusion, I will list below a collection of how this oversight of the structure of the Hilbert space has been repeated throughout the years and throughout journals, including from some of the most distinguished scientists of the field writing in the most distinguished places:

This is from Quantum correlation among photons from a single quantum dot at room temperature. P. Michler, A. İmamoğlu, M. D. Mason, P. J. Carson, G. F. Strouse and S. K. Buratto in Nature 406:968 (2000).: (here they don't refer explicitly to the 1/2)

Michler00b.png

but they do in this text, from A Quantum Dot Single-Photon Turnstile Device. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu and A. İmamoğlu in Science 290:2282 (2000).:

Michler00a-g2.png

This is from Nonclassical radiation from a single self-assembled InAs quantum dot. C. Becher, A. Kiraz, P. Michler, A. İmamoğlu, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang and E. Hu in Phys. Rev. B 63:121312(R) (2001).:

Becher01a-g2.png

This is from Fast recognition of single molecules based on single-event photon statistics. S. Dong, T. Huang an Y. Liu, J. Wang, Liantuan Xiao G. Zhang and S. Jia in Phys. Rev. A 76:063820 (2007).:

Dong07a-g2.png

This is from Photon antibunching from a single lithographically defined InGaAs/GaAs quantum dot. V. B. Verma, M. J. Stevens, K. L. Silverman, N. L. Dias, A. Garg, J. J. Coleman and R. P. Mirin in Opt. Express 19:4182 (2011).:

Verma11a.png

This is from Engineered quantum dot single-photon sources. S. Buckley, K. Rivoire and J. \Vuckovic in Rep. Prog. Phys. 75:126503 (2012).:

Buckley12a-g2.png

the presence of a single quantum emitter can be confirmed by measuring $g^{(2)}(0) < 1/2$.

This is from Quantum Statistics of Surface Plasmon Polaritons in Metallic Stripe Waveguides. G. Di Martino, Y. Sonnefraud, S. K\'ena-Cohen, M. Tame, \c S. K. \"Ozdemir, M. S. Kim and S. A. Maier in Nano Lett. 12:2504 (2012). (supplementary material):

Dimartino12a-g2.png

This is from Subnatural Linewidth Single Photons from a Quantum Dot. C. Matthiesen, A. N. Vamivakas and M. Atatüre in Phys. Rev. Lett. 108:093602 (2012). (supplementary material, they also have the 1/2 lines in the main text but do not comment on their meaning):

Matthiesen12a-g2.png

This is from Resonance in quantum dot fluorescence in a photonic bandgap liquid crystal host. S. G. Lukishova, L. J. Bissell, J. Winkler and C. R. Stroud in Opt. Lett. 37:1259 (2012).

Lukishova12a-g2.png

This is from Bright single-photon sources in bottom-up tailored nanowires. M. E. Reimer, G. Bulgarini, N. Akopian, M. Hocevar, M. Bouwes Bavinck, M. A. Verheijen, E. P.A.M. Bakkers, L. P. Kouwenhoven and V. Zwiller in Nature Comm. 3:737 (2012).:

Reimer13a-g2.png

This is from Single-photon experiments with liquid crystals for quantum science and quantum engineering applications. S. G. Lukishova, A. C. Liapis, L. J. Bissell, G. M. Gehring and R. W. Boyd in Liquid Crystals Reviews 2:111 (2014).

Lukishova14a-g2.png

This is from Evaluation of nitrogen- and silicon-vacancy defect centres as single photon sources in quantum key distribution. M. Leifgen, T. Schröder, F. Gädeke, R. Riemann, V. Métillon, E. Neu, C. Hepp, C. Arend, C. Becher and K. Lauritsen in New J. Phys. 16:023021 (2014).:

Leifgen14a-g2.png

This is from Atomically thin quantum light-emitting diodes. C. Palacios-Berraquero, M. Barbone, D. M. Kara, X. Chen, I. Goykhman, D. Yoon, A. K. Ott, J. Beitner, K. Watanabe, T. Taniguchi, A. C. Ferrari and M. Atatüre in Nature Comm. 7:12978 (2016).:

Palaciosberraquero12a-g2.png

This is from Antibunching and Photon Blockade in a coupled Single Quantum Dot-Cavity System. P. Schwendimann and A. Quattropani in arXiv:1606.04337 (2016).:

Schwendimann-Quattropanni-g2.png

The statement made here (when $g^{(2)}(0)<0.5$ it is highly probable that one photon states are produced) is not technically wrong. References 20, 21 are Photon antibunching from few quantum dots in a cavity. C. Gies, F. Jahnke and W. W. Chow in Phys. Rev. A 91:061804(R) (2015). and Coherent generation of nonclassical light on chip via detuned photon blockade. K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. Sánchez Muñoz, E. del Valle, F. P. Laussy and J. Vučković in Phys. Rev. Lett. 114:233601 (2015). respectively.

This is from Purcell-Enhanced Single-Photon Emission from Nitrogen-Vacancy Centers Coupled to a Tunable Microcavity. H. Kaupp, T. Hümmer, M. Mader, B. Schlederer, J. Benedikter, P. Haeusser, H.-C. Chang, H. Fedder, T. W. Hänsch and D. Hunger in Phys. Rev. Appl. 6:054010 (2016)., who get their own 1/2 criterion (it is not explained how background fluorescence relaxes the condition to 0.7):

Kaupp16a-g2.png

Taking background into account relaxes it to $g^{(2)}\lesssim0.7$.

You also hear it quite a lot in conferences, which is more difficult to reproduce, although you see it in posters too, e.g., this is from Heindel et al. [1]:

Heindel-poster.png

This result, that $g^{(2)}<1/2$ is no guarantee to have less than one particle, even on average (which is a stronger condition than allowing a nonzero probability for n>1 particles) has been publicly announced for the first time in March 2016, at the occasion of the Solid State Quantum Photonics in Sheffield. It is now published[1]. We will keep a list of papers we flag that still use this criterion, apparently sexy enough to have seduced a crowd of experts who got lost at the first corner of the Hilbert space.

  1. Structure of the harmonic oscillator in the space of $n$-particle Glauber correlators. E. Zubizarreta Casalengua, J. C. López Carreño, E. del Valle and F. P. Laussy in J. Math. Phys. 58:062109 (2017). Pdf-48px.png